Subgroup formed by elements of order dividing n

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $G$ be a finite group with identity $1$, and let $n\geq1$ divide its order $|G|$. Define $S_n(G)=\{x\in G:x^n=1\}$, the set of elements whose orders divide $n$. A subgroup is a subset containing $1$ and closed under multiplication and inversion; a characteristic subgroup is preserved by every automorphism of $G$.

Statement recorded in the supplied source.

If $|S_n(G)|=n$, must $S_n(G)$ be a subgroup of $G$? If it is a subgroup, it is automatically characteristic, because its defining equation is invariant under every automorphism. [S2]

Short English statement

Does having exactly n elements whose orders divide n force those elements to form a subgroup, whenever n divides the finite group order?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 3330 (OPG-732). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Open Problem Garden, Subgroup formed by elements of order dividing n, node 732, original problem page. Original question and full discussion. https://www.openproblemgarden.org/op/subgroup_formed_by_elements_of_order_dividing_n.